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Convection in a rotating cylinder. Part 1 Linear theory for moderate Prandtl numbers

Published online by Cambridge University Press:  26 April 2006

H. F. Goldstein ,
E. Knobloch ,
I. Mercader  and
M. Net
H. F. Goldstein
Affiliation:
Department of Physics, University of California, Berkeley, CA 94720, USA
E. Knobloch
Affiliation:
Department of Physics, University of California, Berkeley, CA 94720, USA
I. Mercader
Affiliation:
Departament de Física Aplicada, Universitat Politècnica de Catalunya, E 08034 Barcelona, Spain
M. Net
Affiliation:
Departament de Física Aplicada, Universitat Politècnica de Catalunya, E 08034 Barcelona, Spain
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Abstract

The onset of convection in a uniformly rotating vertical cylinder of height h and radius d heated from below is studied. For non-zero azimuthal wavenumber the instability is a Hopf bifurcation regardless of the Prandtl number of the fluid, and leads to precessing spiral patterns. The patterns typically precess counter to the rotation direction. Two types of modes are distinguished: the fast modes with relatively high precession velocity whose amplitude peaks near the sidewall, and the slow modes whose amplitude peaks near the centre. For aspect ratios τ ≡ d/h of order one or less the fast modes always set in first as the Rayleigh number increases; for larger aspect ratios the slow modes are preferred provided that the rotation rate is sufficiently slow. The precession velocity of the slow modes vanishes as τ → ∞. Thus it is these modes which provide the connection between the results for a finite-aspect-ratio System and the unbounded layer in which the instability is a steady-state one, except in low Prandtl number fluids.

The linear stability problem is solved for several different sets of boundary conditions, and the results compared with recent experiments. Results are presented for Prandtl numbers σ in the range 6.7 ≤ σ ≤ 7.0 as a function of both the rotation rate and the aspect ratio. The results for rigid walls, thermally conducting top and bottom and an insulating sidewall agree well with the measured critical Rayleigh numbers and precession frequencies for water in a τ = 1 cylinder. A conducting sidewall raises the critical Rayleigh number, while free-slip boundary conditions lower it. The difference between the critical Rayleigh numbers with no-slip and free-slip boundaries becomes small for dimensionless rotation rates Ωh2/v ≥ 200, where v is the kinematic viscosity.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 248 , March 1993 , pp. 583 - 604
Copyright
© 1993 Cambridge University Press

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References

Buell, J. C. & Catton, I. 1983a The effect of wall conduction on the stability of fluid in a right circular cylinder heated from below. Trans. ASME C: J. Heat Transfer 105, 255– 260.Google Scholar
Buell, J. C. & Catton, I. 1983b Effect of rotation on the stability of a bounded cylindrical layer of fluid heated from below. Phys. Fluids 26, 892– 896.Google Scholar
Busse, F. H. 1978 Non-linear properties of thermal convection. Rep. Prog. Phys. 41, 1929– 1967.Google Scholar
Catton, I. & Edwards, D. K. 1970 Initiation of thermal convection in finite right circular cylinders. AIChE J. 16, 594– 601.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic. Stability. Dover.
Chossat, P. 1982 Interactions entre bifurcations par brisure partielle de symétrie sphérique. Ann. Sci. Éc. Norm. Sup. 4 (15), 117– 145.Google Scholar
Ecke, R. E., Zhong, F. & Knobloch, E. 1992 Hopf bifurcation with broken reflection symmetry in rotating Rayleigh–Bénard convection. Europhys. Lett. 19, 177– 182.Google Scholar
Goldstein, H. F. & Knobloch, E. 1991 Linear stability of rotating Rayleigh–Bénard convection in a finite cylinder. Bull. Am. Phys. Soc. 36, 2649– 2650.Google Scholar
Homsy, G. M. & Hudson, J. L. 1971 The asymptotic stability of a bounded rotating fluid heated from below: the conductive basic state. J. Fluid Mech. 45, 353– 373.Google Scholar
Jones, C. A. & Moore, D. R. 1979 The stability of axisymmetric convection. Geophys. Astrophys Fluid Dyn. 11. 245– 270.Google Scholar
Joseph, D. D. 1971 Stability of convection in Containers of arbitrary shape. J. Fluid Mech. 47, 257– 282.Google Scholar
Küppers, G. & Lortz, D. 1969 Transition from laminar convection to thermal turbulence in a rotating fluid layer. J. Fluid Mech. 35, 609– 620.Google Scholar
Marqués, F. 1990 On boundary conditions for velocity potentials in confined flows: Application to Couette flow. Phys. Fluids A 2, 729– 737.Google Scholar
Marques, F., Mercader, I., Net, M. & Massaguer, J. M. 1992 Thermal convection in vertical cylinders. I: A method based on potentials of velocity. Comput. Meth. Appl. Mech. Engng (submitted).Google Scholar
Mercader, I., Net, M. & Falques, A. 1991 Spectral methods for high order equations. Comput. Meth. Appl. Mech. Engng 91, 1245– 1251.Google Scholar
Niiler, P. P. & Bisshopp, F. E. 1965 On the influence of Coriolis force on the onset of convection. J. Fluid Mech. 22, 753– 761.Google Scholar
Rand, D. 1982 Dynamics and symmetry. Predictions for modulated waves in rotating fluids. Arch. Rat. Mech. Anal. 79, 1– 37.Google Scholar
Sabry, M. N. 1984 Sur une nouvelle méthode intégrale et son application aux écoulements de fluides visqueux avec ou sans transfert de chaleurs. Ph.D. thesis, Institut National Polytechnique de Grenoble.
Schlüter, A., Lortz, D. & Busse, F. H.1965 On the stability of steady finite amplitude convection. J. Fluid Mech. 23, 129– 144.Google Scholar
Zhong, F., Ecke, R. E. & Steinberg, V. 1991 Asymmetric modes and the transition to vortex structures in rotating Rayleigh-Bénard convection. Phys. Rev. Lett. 67, 2473– 2476.Google Scholar